Class 12

Probability Review

• (General) Addition Rule

• (General) Multiplication Rule

• Dependence vs Independence

The General Addition Rule

The probability of event A or event B occurring is the sum of the probability that A occurs and the probability that B occurs minus the probability that A and B occurs.

\[ \begin{aligned} P(A \operatorname{or} B) &= P(A) + P(B) - P(A \operatorname{and} B) \\ &= P(A) + P(B) - P(A \cup B) \\ &= P(A \cap B) \end{aligned} \]

The Addition Rule for Disjoint Events

When events A and B are disjoint, the probability of event A or event B occurring is just the sum of the probability that A occurs and the probability that B occurs, because the probability that event A and event B occurs is 0.

\[ \begin{aligned} P(A \operatorname{or} B) &= P(A) + P(B) - P(A \operatorname{and} B) \\ &= P(A) + P(B) \\ &= P(A \cap B) \end{aligned} \]

The General Multiplication Rule

The probability of event A and event B occurring is the product of the probability that A occurs and the conditional probability that B occurs given that A has already occurred.

\[ \begin{aligned} P(A \operatorname{and} B) &= P(A) \times P(B \operatorname{given} A) \\ &= P(A) \times P(B | A) \\ &= P(A \cup B) \end{aligned} \]

Independent Processes

• If random process B is independent of random process A, then the probability of random process B does NOT vary based on the outcome of random process A

• i.e. knowing the outcome of A does NOT provide additional information about the probability of B

• Example: When listening to a playlist using a “true shuffle”, the probability that the next song will be by a particular artist does not change based on whether or not the last song played was also by that artist.

Multiplication Rule for Independent Processes

The probability of event A and event B occurring is the product of the probability that A occurs and the probability that B occurs, because the probability of B does not change based on the outcome of A.

\[ \begin{aligned} P(A \operatorname{and} B) &= P(A) \times P(B \operatorname{given} A) \\ &= P(A) \times P(B | A) \\ &= P(A) \times P(B) \\ &= P(A \cup B) \end{aligned} \]

Practice: Swing Voters

• Pew Research survey asked 2,373 randomly sampled registered voters about their…

  • Political affilitation (Democrat, Republican, Independent)

  • Whether they consider themselves a swing voter (Yes, No)

• 35% responded Independent, 23% identified as swing voters, and 11% identified as both

Pratice: Poverty and Language

The American Community Survey (ACS) provides public data each year to give communities demographic information to plan investments and services. The 2010 ACS estimates that…

• 14.6% of Americans live below the poverty line

• 20.7% of Americans speak a language other than English at home

• 31.1% of Americans live below the poverty line or speak a language other than English at home

Chapter 4 Objectives

• Identify and describe the standard normal and normal distributions

• Standardize normal distributions and calculate Z-scores

• Calculate percentiles and exact probabilities

• Apply the 68-95-99.7 Rule

• Read a QQ-Plot (not in book)

The Normal Distribution

Notation

• \(\mu\) (Greek letter mu) represents the mean

• \(\sigma\) (Greek letter sigma) represents the standard deviation of the mean

• \(N(\mu, \sigma)\) stands for a normal distribution with mean \(\mu\) and standard deviation \(\sigma\)

Histograms and Density Curves

Vocabulary scores for 947 seventh-graders. Both histograms and density curves can be helpful in identifying normal distributions.

Example: OkCupid, Heights of Males

Dashed line is self-reported heights by females on OkCupid. Dark purple line is the normal distribution with the same mean and standard deviation. Light purple line is the US average.

The shape of a normal distribution varies by location (mean) and scale (standard deviation)

Changing the mean shifts the “center” of the distribution. Changing the standard deviation alters the “width” of the distribution (i.e. variability).

Standardizing Normal Distributions with Z-Scores

• A Z-score is the number of standard deviations a value falls above (when positive) or below (when negative) the mean of the data

• Z-scores standardize a normal distribution by…

  • Centering the data at 0 by subtracting the mean from each score

  • Scaling the units of the data to 1 by dividing the centered data by the standard deviation

Calculating the Z-Score

\[ \begin{aligned} Z&=\frac{\operatorname{observed value}-\operatorname{mean}}{\operatorname{standard deviation}} \\ &= \frac{x-\mu}{\sigma} \end{aligned} \]

Example: Test Scores

• SAT scores are normally distributed with \(\mu=1500\) and \(\sigma=300\) (\(N(\mu=1500, \sigma = 300)\))

• ACT scores are normally distributed with \(\mu=21\) and \(\sigma=5\) (\(N(21, 5)\))

How do we compare normal distributions with different locations and scales?

Visualizing Z-Scores

Standardizing the data by converting values to Z-scores puts different distributions on the same scale.

The Standard Normal Distribution

• The standard normal distribution is a normal distribution with \(\mu=0\) and \(\sigma=1\) (written \(N(\mu=0, \sigma=1)\))

• Units of the standard normal distribution are standard deviations (Z-scores) (i.e. 1 unit = 1 SD)

• Observations that are 2+ standard deviations from the mean are considered unusual

The 68-95-99.7 Rule

When data is (nearly) normally distributed…

• ~68% of the observations are within 1 standard deviation of the mean (\(\mu \pm \sigma\))

• ~95% of the observations are within 2 standard deviations of the mean (\(\mu \pm 2\sigma\))

• 99.7% of the observations are within 3 standard deviations of the mean (\(\mu \pm 3\sigma\))

The 68-95-99.7 Rule

The 68-95-99.7 Rule describes approximately what proportion of the observations should lie within 1, 2, and 3 standard deviations of the mean respectively, if the data is normally distributed

Example: Test Scores

Proportions, Probabilities, and Percentiles

A percentile is the proportion or percentage of observations that fall below a given value in a distribution.

Probability Density and Cumulative Density Functions

Calculating Percentiles with Z-Score Tables

You can use a Z-Score Table to look up the percentile that corresponds to a particular Z-Score.

Calculating Probabilities with Z-Score Tables

You can use a Z-Score Table to look up the probability of a particular Z-Score.

Example: Discrete Numeric Variables

Sometimes the normal distribution is an acceptable approximation of a discrete numeric variable, but other distributions may be more appropriate.